Boron nitride nanotubes (BNNTs) have been increasingly investigated for use in a wide range of applications due to their unique physicochemical properties including high hydrophobicity, heat and electrical insulation, resistance to oxidation, and hydrogen storage capacity. They are also valued for their possible medical and biomedical applications including drug delivery, use in biomaterials, and neutron capture therapy. Chemical graph theory provides different tools to investigate different properties of nanotubes. Tools like topological invariants are useful to associate an appropriate number with a networks through which we can guess different hidden properties of under consideration network. There are more then 150 topological indices present in history, but no one gives use perfect result in predicting properties of networks. So there is always a room to introduce some new invariants that help us to gain better results. In this paper, we will introduce some new topological indices and polynomials, namely, Maxmin indices and Maxmin polynomials and, calculate results for three different boron nanotubes, boron triangular nanotube BT[p, q], boron‐α nanotube BT(X)[p, q] and boron‐α nanotube BT(Y)[p, q].

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The graph entropy was proposed by Körner in the year 1973 when he was studying the problem of coding in information theory. The foundation of graph entropy is in information theory, but it was demonstrated to be firmly identified with some established and often examined graph-theoretic ideas. For instance, it gives an equal definition to a graph to be flawless, and it can likewise be connected to acquire lower bounds in graph covering problems. The objective of this study is to solve the open problem suggested by Kwun et al. in 2018. In this paper, we study the weighted graph entropy by taking augmented Zagreb edge weight and give bounds of it for regular, connected, bipartite, chemical, unicyclic, etc., graphs. Moreover, we compute the weighted graph entropy of certain nanotubes and plot our results to see dependence of weighted entropy on involved parameters.

Wasserstein distances in optimal transport provides a mathematical tool to measure distances between functions or more general objects. By Wasserstein distances, we define a distance on the moduli space of a class of statistical manifolds. We construct a Riemannian metric of this space and verify that the defined distance can be regarded as the induced distance of the metric.